introduction to artificial intelligence · revised approach •start with a frontier that contains...
TRANSCRIPT
Introduction to Artificial Intelligence
with Python
Artificial Intelligence
Search
O
X X
O X
Knowledge
P → Q
QP
Uncertainty
Optimization
LearningInbox
Spam
Neural Networks
Language
NP
NP PP
ADJ N P N
artificial
intelligence
with
python
Search
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Search Problems
agent
entity that perceives its environment and acts upon that environment
state
a configuration of the agent and its environment
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initial state
the state in which the agent begins
initial state 5 742
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actions
choices that can be made in a state
actions
ACTIONS(s) returns the set of actions that can be executed in state s
actions1 2
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transition model
a description of what state results from performing any applicable action in any state
transition model
RESULT(s, a) returns the state resulting from performing action a in state s
RESULT( , ) =
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RESULT( , ) =
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transition model
RESULT( , ) =
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state space
the set of all states reachable from the initial state by any sequence of actions
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goal test
way to determine whether a given state is a goal state
path cost
numerical cost associated with a given path
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Search Problems
• initial state
• actions
• transition model
• goal test
• path cost function
solution
a sequence of actions that leads from the initial state to a goal state
optimal solution
a solution that has the lowest path cost among all solutions
node
a data structure that keeps track of - a state - a parent (node that generated this node) - an action (action applied to parent to get node) - a path cost (from initial state to node)
Approach
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
A
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
A
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
B
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
B
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
C D
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
C D
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
D
E
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
DE
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
D
E
AB
C D
F
Frontier
Find a path from A to E.
• Start with a frontier that contains the initial state.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Expand node, add resulting nodes to the frontier.
D
What could go wrong?
E
AB
C D
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Frontier
Find a path from A to E. A
E
AB
C D
F
Frontier
Find a path from A to E.
A
E
AB
C D
F
Frontier
Find a path from A to E.
B
E
AB
C D
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Frontier
Find a path from A to E.
B
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Frontier
Find a path from A to E. A
C D
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AB
C D
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Frontier
Find a path from A to E.
A C D
E
AB
C D
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Frontier
Find a path from A to E.
C D
Revised Approach
• Start with a frontier that contains the initial state.
• Start with an empty explored set.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Add the node to the explored set.
• Expand node, add resulting nodes to the frontier if they aren't already in the frontier or the explored set.
Revised Approach
• Start with a frontier that contains the initial state.
• Start with an empty explored set.
• Repeat:
• If the frontier is empty, then no solution.
• Remove a node from the frontier.
• If node contains goal state, return the solution.
• Add the node to the explored set.
• Expand node, add resulting nodes to the frontier if they aren't already in the frontier or the explored set.
stack
last-in first-out data type
E
AB
C D
F
Frontier
Find a path from A to E. A
Explored Set
E
AB
C D
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Frontier
Find a path from A to E.
A
Explored Set
E
AB
C D
F
Frontier
Find a path from A to E.
Explored Set
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A
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Frontier
Find a path from A to E.
Explored Set
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Explored Set
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Frontier
Find a path from A to E.
Explored Set
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C D
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Frontier
Find a path from A to E.
Explored Set
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Frontier
Find a path from A to E.
Explored Set
C F
BA D
E
AB
C D
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Frontier
Find a path from A to E.
Explored Set
C
BA D F
E
AB
C D
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Frontier
Find a path from A to E.
Explored Set
E
BA D F C
E
AB
C D
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Frontier
Find a path from A to E.
Explored Set
E
BA D F C
E
AB
C D
F
Frontier
Find a path from A to E.
Explored Set
BA D F C
Depth-First Search
depth-first search
search algorithm that always expands the deepest node in the frontier
Breadth-First Search
breadth-first search
search algorithm that always expands the shallowest node in the frontier
queue
first-in first-out data type
E
AB
C D
F
Frontier
Find a path from A to E. A
Explored Set
E
AB
C D
F
Frontier
Find a path from A to E.
Explored Set
A
E
AB
C D
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Frontier
Find a path from A to E.
Explored Set
A
B
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C D
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Frontier
Find a path from A to E.
Explored Set
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B
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Frontier
Find a path from A to E.
Explored Set
A B
C D
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AB
C D
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Frontier
Find a path from A to E.
Explored Set
A B
C D
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AB
C D
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Frontier
Find a path from A to E.
Explored Set
A B C
D
E
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AB
C D
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Frontier
Find a path from A to E.
Explored Set
A B C
D E
E
AB
C D
F
Frontier
Find a path from A to E.
Explored Set
A B C D
E
F
E
AB
C D
F
Frontier
Find a path from A to E.
Explored Set
A B C D
E F
E
AB
C D
F
Frontier
Find a path from A to E.
Explored Set
A B C D
F
B
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Depth-First Search
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Depth-First Search
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Breadth-First Search
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Breadth-First Search
uninformed search
search strategy that uses no problem-specific knowledge
informed search
search strategy that uses problem-specific knowledge to find solutions more efficiently
greedy best-first search
search algorithm that expands the node that is closest to the goal, as estimated by a heuristic function h(n)
B
A
Heuristic function?
B
D
C
A
Heuristic function?
B
D
C
A
Heuristic function? Manhattan distance.
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
11 9 7 3 2 B
12 10 8 7 6 4 1
13 12 11 9 7 6 5 2
13 10 8 6 3
14 13 12 11 9 7 6 5 4
13 10
A 16 15 14 11 10 9 8 7 6
Greedy Best-First Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 12 10 9 8 7 6 4
13 11 5
A 16 15 14 12 11 10 9 8 7 6
Greedy Best-First Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 12 10 9 8 7 6 4
13 11 5
A 16 15 14 12 11 10 9 8 7 6
Greedy Best-First Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 12 10 9 8 7 6 4
13 11 5
A 16 15 14 12 11 10 9 8 7 6
Greedy Best-First Search
A* search
search algorithm that expands node with lowest value of g(n) + h(n)
g(n) = cost to reach node h(n) = estimated cost to goal
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 12 10 9 8 7 6 4
13 11 5
A 16 15 14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 12 10 9 8 7 6 4
13 11 5
A 1+16 15 14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 12 10 9 8 7 6 4
13 11 5
A 1+16 2+15 14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 12 10 9 8 7 6 4
13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 10 9 8 7 6 5 4 2
13 6+11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 9 8 7 6 5 4 2
13 6+11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 8 7 6 5 4 2
13 6+11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 7 6 5 4 2
13 6+11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 6 5 4 2
13 6+11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 11+6 5 4 2
13 6+11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 11+6 12+5 4 2
13 6+11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
13 6+11 5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
13 6+11 14+5 3
14 13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
13 6+11 14+5 3
14 6+13 5+12 10 9 8 7 6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
13 6+11 14+5 3
14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
10 9 8 7 6 5 4 3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 9 8 7 6 5 4 3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 8 7 6 5 4 3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 13+8 7 6 5 4 3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 13+8 14+7 6 5 4 3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 13+8 14+7 15+6 5 4 3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 13+8 14+7 15+6 16+5 4 3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 13+8 14+7 15+6 16+5 17+4 3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 13+8 14+7 15+6 16+5 17+4 18+3 2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 13+8 14+7 15+6 16+5 17+4 18+3 19+2 1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
11+10 12+9 13+8 14+7 15+6 16+5 17+4 18+3 19+2 20+1 B
10+11 1
9+12 7+10 8+9 9+8 10+7 11+6 12+5 13+4 2
8+13 6+11 14+5 3
7+14 6+13 5+12 10 9 8 7 15+6 4
4+13 11 5
A 1+16 2+15 3+14 12 11 10 9 8 7 6
A* Search
A* search
optimal if - h(n) is admissible (never overestimates the
true cost), and - h(n) is consistent (for every node n and
successor n' with step cost c, h(n) ≤ h(n') + c)
Adversarial Search
XO X
OXOX
XO X
OXOX
Minimax
O X X
O O
O X X
X O X
O O X
X X O
O X
X O
X O X
1-1 0
Minimax
•MAX (X) aims to maximize score.
•MIN (O) aims to minimize score.
Game
• S0 : initial state
• PLAYER(s) : returns which player to move in state s
• ACTIONS(s) : returns legal moves in state s
• RESULT(s, a) : returns state after action a taken in state s
• TERMINAL(s) : checks if state s is a terminal state
• UTILITY(s) : final numerical value for terminal state s
Initial State
PLAYER(s)
PLAYER( ) = X
XPLAYER( ) = O
ACTIONS(s)
X O
O X X
X OACTIONS( ) = { , }
O
O
RESULT(s, a)
X O
O X X
X ORESULT( , ) =O
O X O
O X X
X O
TERMINAL(s)
O
O X
X O XTERMINAL( ) = false
O X
O X
X O XTERMINAL( ) = true
UTILITY(s)
O X
O X
X O XUTILITY( ) = 1
O X X
X O
O X OUTILITY( ) = -1
VALUE: 1
O X OO X XX X O
X OO X XX O
PLAYER(s) = O
O X OO X XX O
X OO X XX O O
MIN-VALUE:0
O X OO X XX X O
X X OO X XX O O
VALUE:0
VALUE:1
MAX-VALUE:1
MAX-VALUE:0
X OO X XX O
O X OO X XX O
X OO X XX O O
MIN-VALUE:0
O X OO X XX X O
X X OO X XX O O
VALUE:0
VALUE:1
MAX-VALUE:1
MAX-VALUE:0
PLAYER(s) = O
PLAYER(s) = X
X O
O X XX O
O X O
O X XX O
X O
O X XX O O
MIN-VALUE:0
O X O
O X XX X O
X X O
O X XX O O
VALUE:0
VALUE:1
MAX-VALUE:1
MAX-VALUE:0
X OO XX O
X X O
O XX O
X X O
O X OX O
X X O
O XX O O
MIN-VALUE:-1
X X O
O X XX O O
VALUE:0
VALUE:-1
MAX-VALUE:0
X O
O XX X O
VALUE:1
MAX-VALUE:1
9
5 3 9
9
5 3 9 2 8
8
8
Minimax
• Given a state s:
•MAX picks action a in ACTIONS(s) that produces highest value of MIN-VALUE(RESULT(s, a))
•MIN picks action a in ACTIONS(s) that produces smallest value of MAX-VALUE(RESULT(s, a))
Minimax
function MAX-VALUE(state): if TERMINAL(state): return UTILITY(state) v = -∞ for action in ACTIONS(state): v = MAX(v, MIN-VALUE(RESULT(state, action))) return v
Minimax
function MIN-VALUE(state): if TERMINAL(state): return UTILITY(state) v = ∞ for action in ACTIONS(state): v = MIN(v, MAX-VALUE(RESULT(state, action))) return v
Optimizations
4
5 2
2 64
3
9 73
4
4 58
4
5 ≤2
2
≤3
9 3
4
4 58
Alpha-Beta Pruning
255,168total possible Tic-Tac-Toe games
288,000,000,000
total possible chess gamesafter four moves each
total possible chess games(lower bound)
2900010
Depth-Limited Minimax
evaluation function
function that estimates the expected utility of the game from a given state
https://xkcd.com/832/
Search
Introduction to Artificial Intelligence
with Python